Aristarchus and the heliocentric theory

The earth has orbited around the sun since 1609. At least that’s when Kepler’s book on the subject came out, Astronomia nova (‘The new astronomy’). Copernicus had proposed a heliocentric theory in 1543, but with circular orbits it was a lousy model. The geocentric Ptolemaic system continued to be the better model of planetary motion until Kepler came along.

But there was another precedent. Sometime around 280 BCE, in ancient Greece, Aristarchus of Samos proposed a heliocentric model. What exactly did Aristarchus argue? How did he arrive at his theory, what did people think of it, and why did it end up being neglected?

Contemplating the moon (AI generated)

The last question is the simplest: Aristarchus’ theory was neglected because his writings on the subject were lost. Also, other ancient astronomers found that geocentrism, with epicycles, produced a superior model of planetary motion — and they were right. Even though the reason they were right had nothing to do with the planets’ real motion, and everything to do with a form of mathematical analysis that wasn’t fully developed until the 1800s.

Aristarchus of Samos

We don’t know much about Aristarchus’ life. He was born on the island of Samos, probably in the 310s BCE, a decade or two after Alexander’s death. What we know of his dates comes from just three facts:

• We’re told he studied under Straton of Lampsakos, who was the head of the Peripatos in Athens from 287 until 269 BCE.
• Aristarchus observed the summer solstice in 280 BCE.
• His heliocentric model was discussed by Archimedes in the 240s or 230s BCE.

So we know he spent a period in Athens at some point, but nothing else about his movements. We know he developed a heliocentric theory; he measured the sizes and distances of the moon and sun; discovered some trigonometric inequalities; and invented two instruments, something called the ‘disc on a level surface’, and the skaphe, a bowl with a fixed needle and gauge markings for measuring the sun’s position.

Map showing Samos (base image: Google Earth)

Much more precise equipment for measuring the sun’s position was developed pretty soon afterwards. But Aristarchus’ skaphe was straightforward enough to stick around: three centuries later, Pliny reports a bunch of skaphe readings of the sun’s altitude, varying depending on how far north you are.

Note. Pliny, Natural history 2.74, with a description of the skaphe. Aristarchus inventing the skaphe and the discum in planitia: Vitruvius 9.8.1. More precise devices for measuring the sun’s altitude are described by Ptolemy, Almagest 1.12. One of them, a device consisting of two concentric vertical rings, was in use in Meroë, Sudan, by the 2nd century BCE, and is probably also the device used by Eratosthenes: see here for details.

Only one book by Aristarchus survives: On the sizes and distances of sun and moon. It isn’t widely read. In it he measures the moon as being considerably smaller than the earth, and the sun as much bigger. This is also where we find him inventing some bits of trigonometry from first principles.

Note. Edition of Aristarchus’ On the sizes: Heath 1913: 317–414; text and translation at 352–411. Reader beware: Heath’s introduction is seriously marred by relying on Hultsch’s botched reckoning of two ancient distance units, the Egyptian schoinos and Greek stadion. See here. For a more recent and accurate discussion, see Berggren and Sidoli 2007.

The heliocentric theory and On the sizes

The heliocentric theory isn’t mentioned in Aristarchus’ surviving book. We have to rely on other ancient reports — and they aren’t generous with details.

Our main source is Archimedes. He brings up the heliocentric theory in a mathematical exercise, about devising a numerical notation capable of representing very large quantities.

Aristarchus of Samos, however, published writings of certain propositions, where it appears from the premises that the cosmos is many times larger than the standard [i.e. geocentric] cosmos. His suggestion is that the fixed stars and sun remain motionless, and the earth orbits around the sun in a circle, the sun at the centre of its path; and the sphere of fixed stars lies around the sun, with the sun at its centre. And its size [i.e. the sphere of fixed stars] is such that the circle of the earth's orbit has the same proportion to the distance of the fixed stars, as the centre of the sphere has to its surface.

This is obviously impossible: the centre of the sphere has no size, so it has to be understood as having no ratio to the sphere’s surface. So we take Aristarchus’ meaning to be: we suppose that the earth [in the geocentric model] is analogous to the centre of the cosmos [in the heliocentric model]; therefore, the earth’s ratio to the cosmos as we imagine it [i.e. geocentric] is the same as the ratio of the sphere on which the circle of the earth's orbit is inscribed to the sphere of fixed stars [in the heliocentric model].

Archimedes, Sand-reckoner 4–6 (ii.218 Heiberg)

This is obscurely phrased. Essentially, the second paragraph is saying that a lower bound for the size of a heliocentric cosmos has to be vastly larger than that of a geocentric cosmos.

Archimedes doesn’t say why. Presumably because of the parallax problem: as the earth moves around the sun, the fixed stars ought to shift their parallax in a yearly cycle. But they don’t. Therefore, either the earth doesn’t go around the sun, or the fixed stars are enormously more distant than the geocentric model would require.

(Archimedes goes on to work out how many grains of sand it would take to fill a very large cosmos. He calculates a lower bound for the universe’s diameter of a little under 2 light years — or rather, 100 trillion stadia — with room for 10⁶³ grains of sand.)

Still a bit of counting to do (AI generated)

This tells us: (1) Aristarchus proposed a heliocentric model; (2) he appreciated that the distance to the fixed stars has to be treated as effectively infinite. But it doesn’t tell us why Aristarchus thought this was better than the conventional geocentric model.

Two other passages in Plutarch are worth noting, dating to the 2nd century CE. One states that there were only two notable heliocentrists, Aristarchus and Seleucus; and that Aristarchus’ heliocentric model was only a proposal, while Seleucus regarded it as evidently true. The other passage tells a story of Aristarchus having a clash with a Stoic philosopher, Cleanthes. Several other sources discuss whether the earth is in motion rotating on its axis; the most prestigious figures, like Aristotle, Hipparchus, and Ptolemy, conclude it’s the sky that rotates.

Note. Plutarch, Platonic questions 1006c; Plutarch, On the face in the circle of the moon 922f–923a. On the question of Seleucus’ exact contribution, and the meaning of Plutarch’s word ἀποφαινόμενος, see Neugebauer 1975.ii: 697–698. On the earth’s rotation, see Aristotle, On the sky 296a–b (against); Heracleides of Pontus frs. 104–117 Wehrli (in favour); Seneca, Natural questions 7.2.3 (agnostic).

These still don’t tell us why Aristarchus favoured the heliocentric theory. But Aristarchus’ surviving work, On the sizes and distances of sun and moon, gives a pretty broad hint.

Aristarchus calculated the moon to have a diameter equivalent to about one third of an earth diameter, and the sun, about seven earth diameters. He based this on observations of the size of the earth’s shadow on the moon during lunar eclipses, and the angle between the sun and moon when the moon is half full.

His final figures are way off, because his observational tools ... well, sucked. He also wrongly assumed the moon subtends an angle of 2° as seen from earth, when it’s actually 0.5°. (And no, this isn’t a result of a typographical ambiguity.) Also, he couldn’t use trig functions on a modern calculator to convert angle measurements to distances, so he had to discover his trigonometrical inequalities to obtain lower and upper bounds.

An excerpt of Aristarchus’ On the sizes (Heath 1913: 364–365). At the bottom is where he mistakes the angular size of the moon: ‘And since it is assumed that the moon subtends a 15th of a zodiacal sign ...’ (where each zodiacal sign occupies a 12th of a circle, or 30°).

In other respects, his calculations are good. And he got one essential point right: the sun is much bigger than the earth. Sure, his figure for the sun’s size is missing a couple of zeroes. But it may still have been enough for him to infer that the cosmos ought to be imagined as centred on the colossal sun, not the puny earth.

For reference, here are his results, along with the actual figures as determined by modern astronomy.

	Aristarchus	Actual	Actual (km)
lunar diameter	0.3167 to 0.3981 earth diameters	0.2727 earth diameters	3475 km
lunar distance	22.50 to 30.00 lunar diameters	110.6 lunar diameters	384,400 km
solar diameter	6.333 to 7.167 earth diameters	109.2 earth diameters	1,392,000 km
solar distance	18.00 to 20.00 lunar distances	389.2 lunar distances	149,600,000 km

Note. Numbers are given to 4 s.f. The ‘actual’ columns show averages. For Aristarchus’ numbers see Heath 1913: 338. We don’t know what Aristarchus reckoned for the earth’s diameter: he may perhaps have known the 300,000 stadia estimate for the circumference that Archimedes mentions.

On the point of comparative sizes, he was obviously right, and everyone knew it. That may have been enough to push him towards the heliocentric theory. And if he realised that the theory could also explain the retrograde motion of the planets — well, that may have been a nice bonus.

What people thought of Aristarchus’ theory

People didn’t really take to heliocentrism. As I mentioned, we know of only one other ancient heliocentrist by name, Seleucus (mid-2nd century BCE).

Note. Seleucus: from the Erythraean Sea according to Strabo 3.5.9, Diels DG 328.5; from Seleuceia according to Strabo 16.1.6. He argued that Ocean tides are related to the motion of the moon (Strabo 1.1.9, 3.5.9); like Heracleides and Aristarchus he argued that the universe is infinite (Diels DG 328.5; a 10th century report quoted and translated by Pines 1963: 197).

Aristotle offered three separate objections to the idea that the earth orbits the sun — and he did so several decades before Aristarchus came along (On the sky 296a–296b).

1. If the earth were in motion, either that motion must be the result of a force acting on the earth, in which case it’s non-natural and temporary; or it must be a natural motion shared by all parts of the earth, in which case objects should hover relative to the earth’s surface, but they don’t.
2. If the earth orbited around something else we would observe stellar parallax, but we don’t.
3. Weight falls to the ground: that is, it has a natural motion towards the centre of the earth. A natural phenomenon must be universal. Therefore this must actually be motion towards the centre of the cosmos. The fact that the earth’s centre is also in the same place is simply a result of the earth itself gravitating towards the centre.

Points 1 and 3 come from the observed fact that the earth is spherical. Point 2 is more specifically astronomical. We don’t know how Aristarchus would have responded to points 1 and 3; his solution to point 2 was to posit that the universe is infinite, or effectively infinite. Tycho Brahe too, in the 1600s, thought point 2 was heliocentrism’s weak point. Ptolemy thought it was point 3 (Almagest 1.7 = 24-25 Heiberg).

It’s kind of amazing how Aristotle’s points are completely wrong — but to see that they’re wrong, you need another two thousand years of science. Aristotle’s first and third objections weren’t resolved until the publication of Newton’s laws in 1687. Stellar parallax wasn’t measured until the 1830s. He was wrong ... but what a way to be wrong!

There’s no reason to imagine anyone ever thought the heliocentric theory violated any taboos. Writers like Archimedes and Ptolemy were happy to take it seriously and consider its implications, even if they disagreed with it.

The main evidence of active opposition to the heliocentric theory relates to the Stoic philosopher Cleanthes. We know Cleanthes wrote a tract called Against Aristarchus (Diogenes Laertius 7.174). Plutarch has an anecdote of Cleanthes saying that Aristarchus ought to be charged with ‘impiety’ (asebeia): in context it’s clear that that’s just a hyperbolic joke. But his opposition to the theory was genuine.

(Pharnaces said,) ‘You won’t induce me to give an account of what you’re accusing the Stoics of, until I get an account from you for turning the cosmos upside down!’

Then Lucius laughed and said, ‘Just don’t bring a charge of impiety against us! — like when Cleanthes thought the Greeks should accuse Aristarchus of Samos of impiety, because he was disturbing the foundation of the cosmos, trying to preserve observations by suggesting that the sky is immobile, and the earth orbits along the ecliptic, and rotates on its own axis.’

Plutarch, On the face in the circle of the moon 922f–923a

Note. Plutarch’s phrasing ‘preserve observations’ (φαινόμενα σῴζειν) is standard; it also appears in Simplicius’ account of Heracleides’ theory that the earth rotates (519,9–11 ed. Heiberg = Heracleides fr. 108 Wehrli). Russo and Medaglia 1996 prefer the manuscript reading Ἀρίσταρχος ... Κλεάνθη to the usual emendation Ἀρίσταρχον ... Κλεάνθης, so that Aristarchus accuses Cleanthes of impiety rather than the other way round. However, the person in the accusative case here (a) is the one being accused, (b) is from Samos, and (c) is ‘disturbing the foundation of the cosmos’. These are very easily ascribed to Aristarchus; they cannot possibly be ascribed to Cleanthes. Cleanthes (a) wrote a treatise called Against Aristarchus, (b) was from Assos, and (c) was no astronomer. Russo and Medaglia have to make multiple other emendations to get the sentence to make sense, and their emendations rely on speculations about Cleanthes’ teachings. The usual emendation is far more robust.

Cleanthes — a boxer in his youth — ready to throw hands over the heliocentric model (AI generated)

Aristarchus’ theory wasn’t suppressed, it was just abandoned. The weight of opinion was against it. There was Cleanthes’ treatise; and actual astronomers had a more effective model to work with. Heath suggests that it was Hipparchus’ opposition, in the 2nd century BCE, that ‘sealed the fate of the heliocentric hypothesis’ (1913: 308). Certainly the extant Hipparchan-Ptolemaic system, with its eccentric orbits and epicycles, is better at modelling the motion of the planets.

People often scoff at the idea of Ptolemaic epicycles, but they’re missing the point. Epicycles are incredibly effective because each one is a term in a Fourier series.

A simplified form of the Ptolemaic model, ignoring eccentricity (base image: Youtube)

The principle of Fourier analysis is that any periodic function can be modelled as a composite of simple harmonic motions. Each term in a Fourier series represents a circular motion of a given magnitude and frequency. The more terms, the more accurate the model.

So even though Fourier didn’t formalise the idea until 1822, the Ptolemaic system uses the same principle. Ptolemy represents planetary motion as a Fourier series with coefficients determined by trial and error. The first term in the series is the deferent, the second is the epicycle.

No one in the present day objects when an mp3 compresses sound using 1024 epicycles, instead of encoding information about pitches, timbre, and instrumentation. It isn’t physically real, but it’s a very effective model. Epicycles in the Ptolemaic system work exactly the same way.

Approximations of a square wave using Fourier series of 5 terms, 10 terms, and 125 terms (base image: Youtube).

Aristarchus was nearly forgotten by the time Copernicus reintroduced the heliocentric model in the early 1500s. The main source for Aristarchus’ theory, Archimedes’ Sand-reckoner, didn’t appear in print until 1544, the year after Copernicus’ death. Copernicus’ manuscript of De revolutionibus did refer to Aristarchus’ heliocentric theory, but he clearly didn’t know much about it. He removed the reference in the print edition.

Note. First print edition of the Sand-reckoner: Gechauff 1544: 120–127 (current edition: Heiberg 1913, ≈ 1881: 242–291). Copernicus’ manuscript: Biblioteka Jagiellońska, BJ Rkp. 10000 III, at f. 11v: ‘It is feasible that, for these and similar reasons, Philolaus [the Pythagorean] perceived that the earth is mobile; several sources report that Aristarchus of Samos was of the same view, for some reason other than that which Aristotle cites and refutes.’ The print edition mentions Aristarchus only in a separate context (1543: 65v).

References

• Berggren, J. L.; Sidoli, N. 2007. ‘Aristarchus’s On the sizes and distances of the sun and moon: Greek and Arabic texts.’ Archive for history of exact sciences 61: 213–254. [Sci-hub]
• Copernicus, N. 1543. De revolutionibus orbium coelestium. Nürnberg. [Internet Archive]
• Gechauff, Th. (alias Venatorius) 1544. Ἀρχιμήδους τοῦ Συρακουσίου, τὰ μεχρὶ νῦν σωζόμενα, ἅπαντα. Archimedis Syracusani philosophi ac geometrae excellentissimi opera. Basel. [Google Books]
• Heath, T. L. 1913. Aristarchus of Samos. The ancient Copernicus. Oxford. [Internet Archive]
• Heiberg, J. L. 1913. Archimedes opera omnia, 2nd edition (first publ. 1881) vol. 2. Leipzig. [1881 edition: Internet Archive]
• Neugebauer, O. 1975. A history of mathematical astronomy, 2 vols. Berlin/Heidelberg.
• Pines, S. 1963. ‘Un fragment de Séleucus de Séleucie conservé en version arabe.’ Revue d’histoire des sciences 16: 193–209. [JSTOR]
• Russo, L.; Medaglia, S. M. 1996. ‘Sulla presunta accusa di empietà ad Aristarco di Samo.’ Quaderni Urbinati di cultura classica 53: 113–121. [JSTOR]